#20477: reduced words in complex reflection group
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Reporter: stumpc5 | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-7.2
Component: combinatorics | Resolution:
Keywords: complex reflection groups, | Merged in:
reduced words | Reviewers:
Authors: | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: |
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Comment (by nthiery):
Just wondering: is that still the case for a complex reflection group W
that, if you take a parabolic subgroup W_I and its coset representatives
W^I the factorization of an element w=w_I w^I is reduced?
(that is len(w)=len(w_I)+len(w^I))?
If yes, would there be a way to compute this decomposition? Then one could
use induction to compute the reduced word for w_I and only do the depth
first search on W^I?
I would assume that, if one chooses carefully the base for the permutation
group (e.g. by starting with the simple roots s_i for i not in I, and then
the others), then W_I would be one of the groups in the stabilizer chain.
So expressing w in terms of the strong generators would give the
decomposition.
It's probably best to take I a maximal parabolic subgroup. Then, W_I
should be the first subgroup in the stabilizer chain, and the strong
generators for this last inclusion W_I \subset W should be exactly the
coset representatives. And the Shreier tree computed by GAP might actually
just be the depth first search tree on then (I don't know if GAP uses a
depth first search though).
All of this to be taken with a grain of salt ...
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Ticket URL: <http://trac.sagemath.org/ticket/20477#comment:3>
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